## Tuesday, January 31, 2017

### miscellaneous problem from differential equations

miscellaneous problem from differential equations
find the equation of the curve passing through (0,pi/4) whose differential equation is
sinx cosy dx +  cosx siny dy =0

here cosy with dx and cosx with dy should be removed.

So divide each term with cosy cosx

the resulting equation is of variable separable type.

integrate term by term

Now to get the value of C, use the given condition that the curve passes through  (0,pi/4)

put x =0  and y = pi/4 and solve for C.

Variable separable differential equation
* show that the general solution of y' + {[ (y^2)+y + 1]/[x^2+x+1] = 0
is given by x+y+1=A[1-x-y-2xy]

*solve y e^(x/y) = [ y e^(x/y) + (y^2) ]dy  [ y is not equal to 0]
solution of differential equation which is not homogeneous but which can be solved using x=vy

solution of a second order differential equation using reduction of order
solve y"-y = 0 if y = coshx is one of the solutions
using the formula for reduction of order
solution of solution of a second order differential equation using reduction of order

### variation of parameter method

solve xy" - 4y' = x^4 by method of variation of parameter
solution to problem on differential questions using variation of parameter method

### orthogonal trajectory of y(1+x ² ) = Cx

find the orthogonal trajectory of y(1+x ² ) = Cx
answer to problem on  orthogonal trajectory of y(1+x ² ) = Cx

### orthogonal trajectory of y = (k/x)

find the orthogonal trajectory of y = (k/x)
solution to  find the orthogonal trajectory of y = (k/x)

formulae on integration

PAGE 1 BASIC INTEGRATION

PAGE 2 INTEGRATION BY SUBSTITUTION

PAGE 3 INTEGRATION BY COMPLETION OF SQUARES

PAGE 4 INTEGRATION BY PARTS

PAGE 5 INTEGRATION BY MANIPULATION OF NUMERATOR IN TERMS OF DENOMINATOR

PAGE 6 INTEGRATION USING PARTIAL FRACTIONS

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